## Abstract

The lateral shifts (i.e., the spatial GH shifts) of reflected wave from a uniaxial anisotropic chiral metamaterial slab are predicted based on the stationary-phase method. The two cases of the uniaxial chiral slab are investigated in detail. It is found that the negative uniaxial chiral slab can be realized for both the two cases as the group refraction of the left circularly polarized (LCP) wave is negative. The GH shifts of the perpendicular component (${\Delta}_{sp}$) are similar for both of the cases, which are enhanced at close-to-grazing incidence. However, the GH shifts of the parallel component (${\Delta}_{pp}$) can be enhanced near the Brewster angle. For case (I), by introducing a different chirality parameter and the thickness of the chiral slab, the transition between the positively and negatively enhanced GH shifts of the parallel component (${\Delta}_{pp}$) can be realized. For case (II), the magnitude and the position of the enhancement of the positively enhanced GH shifts (${\Delta}_{pp}$) can be tunable by adjusting the chirality and the thickness of the chiral slab.

© 2017 Optical Society of America

## 1. Introduction

We know that Goos-Hӓnchen (GH) effect refers to the reflected beam experiences a lateral shift in the incident plane from the position predicted by geometrical optics, as a light beam is totally reflected from a planar interface [1]. Recently, the extensive investigations on the lateral shifts have been done due to the potential applications in optical devices. For instance, the enhancement and tunable of GH shifts can be realized in different materials or structures, including photonic crystals [2, 3], periodic structures [4], antiferromagnet [5], absorptive media [6–8], and so on. Apart from the constant spatial GH shifts mentioned before, the light beam also experiences an angular GH shift, which can be theoretically and experimentally observed via the weak measurement scheme [9, 10]. Meanwhile, with the help of the weak measurements, the spatial and angular GH shifts can be observed in total internal reflection (TIR) [11] or in partial reflection [12, 13], Moreover, the spatial and angular GH shifts can also be observed from a lossy surface [14] or in a graphene [15, 16].

On the other hand, since the emergence of left-handed materials (LHMs), which has brought about great opportunities and sophisticated pathways to manipulate light. Thus, the lateral shifts (i.e., the spatial GH shifts) associated with metamaterials with negative refraction [17–19] are of great interest. Moreover, as the important candidate of the metamaterials with negative refraction [20], the chiral metamaterials have received great attention from both theoretical and experimental views [21–24]. Correspondingly, the lateral shifts of the isotropic chiral metamaterials have been studied, which predicted the large positive and negative lateral shifts can be obtained from an isotropic chiral metamaterial [25–27]. On the other hand, the isotropic metamaterials are very difficult to be realized in practice, and the most of the artificial structures are actually anisotropic with respect to the directions of incident waves. Thus with the research going on, the studies of the lateral shifts have been done from the isotropic metamateirals to anisotropic metamaterials [28, 29]. For example, the beam shifting of a Gaussian beam has been analyzed for an anisotropic negative refractive slab in detail [30]. Meanwhile, the thickness and physical parameters of the anisotropic metamaterial slab, as well as the incident angle of the light beam, strongly affect the properties of the lateral shifts [31–33].

In addition, in the previous studies, the lateral shifts of the anisotropic chiral metamaterial have rarely been reported, though the spatial and angle transverse shifts for a Gaussian incident beam have been investigated in Ref [34]. And by investigating the refractive index of the uniaxial chiral media [35, 36], the results show that the anisotropic chiral metamaterials are quite easy to be realized artificially, since the condition to realize negative refraction can be quite loose. Moreover, the optical properties of the uniaxial chiral slab and the guided modes of the uniaxial chiral waveguide have also been studied [37, 38], which predicted the anisotropic chiral metamaterials have more potential applications in optical devices. Taking this into account, in this paper, we would like to investigate the GH shifts of the reflected beam from an anisotropic chiral metamaterial slab with absorption. Generally, the spatial and angle GH shifts occur simultaneously. But for a collimated incident beam, the spatial shifts are dominant, for a focused beam, the angle shifts are dominant. That is to say for a well collimated Gaussian beam, the angular deviation vanishes in the geometrical optics limit. Thus, for simplicity, we consider a monochromatic plane wave, and the spatial GH shifts (i.e., the lateral shifts) are investigated without considering the angle GH shifts. The influences of the parameters of the anisotropic chiral metamaterial will be discussed in detail. The modulation of the enhanced beam shift with positive or negative values and the transition between them can also be obtained.

## 2. Formulation

The configuration for the uniaxial anisotropic chiral metamaterial slab with the thickness *d* is shown in Fig. 1, which the optical axial is perpendicular to the interface. In general, time dependence $\mathrm{exp}\left(-i\omega t\right)$ is applied and suppressed. The constitutive relations of the uniaxial chiral slab are defined as [39]

*ε*

_{0}and

*μ*

_{0}are the permittivity and permeability in vacuum.

*ε*(

_{t}*μ*) and

_{t}*ε*(

_{z}*μ*) are the relative permittivity (permeability) of the uniaxial chiral medium perpendicular and parallel to the optical axial.

_{z}*κ*is the chirality parameter and describes electromagnetic coupling.

Here, we assume that a polarized wave is incident at an angle *θ _{i}* upon the surface of a uniaxial chiral slab. The surface is parallel to the

*xy*plane, and the normal of the interface is along the

*z*direction. In the uniaxial chiral medium, there are two propagation modes: a right circularly polarized (RCP) wave with wave number

*k*and a left circularly polarized (LCP) wave with wave number

_{R}*k*, which are generated as [37]

_{L}*k*), reflected (

_{i}*k*) and transmitted (

_{r}*k*) waves are satisfied ${k}_{t}={k}_{r}={k}_{i}={k}_{0}$, ${\theta}_{t}={\theta}_{r}={\theta}_{i}$, ${k}_{zR(zL)}={k}_{R(L)}\mathrm{cos}{\theta}_{R(L)}$. According to the boundary continuity condition (${k}_{yR}={k}_{yL}={k}_{iy}$), we have ${k}_{R}\mathrm{sin}{\theta}_{R}={k}_{L}\mathrm{sin}{\theta}_{L}={k}_{0}\mathrm{sin}{\theta}_{i}$.

_{t}In a chiral material, an electric or magnetic excitation will produce both the electric and magnetic polarizations simultaneously. As a consequence, the reflected wave must be a combination of both perpendicular (s polarized, TE) and parallel (p polarized, TM) components in order to satisfy the boundary conditions, so does the transmitted wave. For the random polarized incident wave, by matching the boundary conditions at the interface $z=0$ and$z=d$, we will have the relations as the following.

According to the cross polarization in the chiral material, the reflected and transmitted coefficients contain the perpendicular and parallel components. Therefore, both the co-polarized and cross-polarized terms are involved in the coefficients. Then, we have

In the above equation, *R _{ab}* and

*T*($ab=ss,ps,sp,pp$) are the reflected and transmitted coefficients,

_{ab}*a*and

*b*correspond to the polarized state of the reflected and incident wave, respectively. Meanwhile, $a=b$ corresponds to the co-polarized coefficients,

*a*corresponds to the cross-polarized coefficients. The arbitrary reflected coefficient can be written as ${R}_{ab}=\left|{R}_{ab}\right|{e}^{i{\varphi}_{ab}}$, ${\varphi}_{ab}$ is the phase of the reflectance.

^{1}bWhat's more, according to the stationary-phase approach, the lateral shift (the spatial GH shift) of the reflected beam can be defined as

## 3. Results and discussions

We are now in a position to present numerical results on the lateral shifts of the uniaxial chiral slab. For numerical calculation, we let ${\mu}_{t}={\mu}_{z}=1$ for simplicity. There will exist four cases for the permittivity of the uniaxial chiral slab:(I) ${\epsilon}_{t}>0,{\epsilon}_{z}>0$; (II) ${\epsilon}_{t}>0,{\epsilon}_{z}<0$; (III) ${\epsilon}_{t}<0,{\epsilon}_{z}>0$; (IV) ${\epsilon}_{t}<0,{\epsilon}_{z}<0$. In the following discussion, a transverse magnetic (TM polarized) wave is injected into the interface of the uniaxial chiral slab with the angle of incidence *θ _{i}*.

First of all, the case (I) is investigated, and taking into account the dissipation of the actual chiral materials in the resonant frequency, the parameters are chosen as ${\epsilon}_{t}=3+0.01i$, ${\epsilon}_{z}=2+0.01i$, $\omega =2\pi \times 10\text{}\text{GHz}$, and *d* = 10mm. Without loss of generality, we consider two types of chiral slab: (1) a weak chirality as$\kappa <\sqrt{{\epsilon}_{z}{\mu}_{z}}$ (κ = 0.1, 0.3, 0.5); (2) a strong chirality $\kappa >\sqrt{{\epsilon}_{z}{\mu}_{z}}$ (κ = 3, 4, 6). Generally, the wave vector and the Poynting vector are not parallel to each other in the anisotropic metamaterial. Therefore, it is quite necessary to introduce the group refraction angle *θ _{gR}* and

*θ*for the anisotropic chiral slab, which can be given as [35]

_{gL}Now let us inspect the phase and group refraction angles for the weak and strong chirality, as shown in Fig. 2. It can be seen that whatever the chirality is weak or strong, there is always no total reflection. For both of RCP and LCP waves, the phase refraction angles (${\theta}_{R}>0,{\theta}_{L}>0$) are positive, so are the group refraction angles (${\theta}_{gR}>0,{\theta}_{gL}>0$) as the chirality is very weak (*κ* = 0.1, 0.3, 0.5). It indicates that the uniaxial anisotropic chiral metamaterial possesses positive refraction index and only supports the forward wave for the weak chirality. When the chirality is strong (*κ* = 3, 4, 6), the phase and group refraction angles are both positive (${\theta}_{R}>0,{\theta}_{gR}>0$) for RCP wave, thus forward wave is always supported. But for LCP wave, the phase and group refraction angles are respectively positive (${\theta}_{L}>0$) and negative (${\theta}_{gL}<0$), which indicates the uniaxial anisotropic chiral metamaterial possesses negative refraction index and can support the backward wave. Then, we conclude that the uniaxial anisotropic chiral slab can be positive or negative only as adjusting the chirality *κ*. From Fig. 2, we also find the chirality parameter has little effect on the phase refraction, but affects the group refraction much. Moreover, for the anisotropic chiral slab with negative refraction, which corresponds to the large chirality (*κ* = 3, 4, 6) as seen in Fig. 2(d), the group refraction angle of LCP wave will decrease with the increase of the chirality. It means that the increase of the chirality leads to the decrease of the negative refraction.

Then, for this type of uniaxial anisotropic chiral metamaterial, the dependence of the reflectivity $|R{|}^{2}$ (a, b) and the GH shifts $\Delta /\lambda $ (c, d) for both perpendicular (subscript sp) and parallel (subscript pp) components on the angle of incidence are shown in Fig. 3. It is easily found that there is a peak in each reflectivity curve for the perpendicular component, at which $|{R}_{sp}{|}^{2}$ reaches the maximum, as seen in Fig. 3(a). As increasing the chirality, the peak of the reflectivity increases (or decreases) for the weak (or strong) chiral slab. Therefore, the Brewster angle does not exist for the perpendicular components. And from Fig. 3(c), the GH shifts will be greatly enhanced at close-to-grazing incidence, the enhancement of the GH shifts (${\Delta}_{sp}/\lambda $) can change from positive to negative as the chirality varies from weak to strong. In contrast, for the parallel component [see Fig. 3(b) and 3(d)], when chirality is very weak (*κ* = 0.1), there exist two dips of the reflectivity curve, at which the reflectivity reaches the minimum magnitude and can be close to zero. Therefore, the two corresponding angles of incidence can be defined as the Brewster angles, ie., ${\theta}_{B1}\approx {45}^{\circ}$ and ${\theta}_{B2}\approx {63.5}^{\circ}$. As a consequence, one expects that the GH shifts are enhanced near the two Brewster angles, which have two negative peaks as seen in Fig. 3(d). Meanwhile, increasing the chirality will lead to the transition of the enhancements from two negative peaks to a positive and a negative one. Here, we note that for the weak chirality, the Brewster angle corresponding to the large angle of incidence will turned to be pseudo-Brewster angle due to the increase of the minimum reflectivity, which is no longer close to zero as the chirality is relatively large (*κ* = 0.5). The enhanced GH shift corresponding to the large angle of incidence also decreases. When chirality becomes strong (*κ* = 3, 4, 6), one dip will be seen, i.e., there exists one Brewster angle. Therefore, the GH shifts can be enhanced near the Brewster angle with a positive peak. From Fig. 3, we conclude that the Brewster angle does not exist in the cross-polarized reflection but exist in the co-polarized reflection. Moreover, increasing the chirality will lead to the transition of the enhancements from two negative peaks to a positive and a negative peaks, eventually become a positive one.

Since the GH shifts of the co-polarized reflection component are more attractive than that of the cross-polarized reflection component, we aim at the case of co-polarized reflection for both weak and strong chirality. The influences of the thickness of the slab on the GH shifts of the parallel components are shown in Fig. 4. The corresponding reflectivity is shown in the insets of Fig. 4. According to Fig. 4(a), the GH shifts can be negatively enhanced at the two Brewster angles at *d* = 10mm for weak chirality, which has also been shown in Fig. 3(b). By increasing the thickness of the slab, the two negative enhancement of the GH shifts can vary to a positive one, then varies to a negative one again as the further increase of the thickness. While for large chirality, there is always a Brewster angle for different thickness of the slab, and the GH shift can be positively enhanced at the Brewster angle at *d* = 10mm. By increasing the thickness, the enhanced GH shift can change from positive to negative, then to positive again. From Fig. 4 we will find by adjusting the thickness of the slab, the transition between the enhanced positive and negative GH shifts can be realized.

Apart from the aforementioned uniaxial chiral slab of case (I), we also consider the case (II), the corresponding parameters are chosen as ${\epsilon}_{t}=3+0.01i$, ${\epsilon}_{z}=-4+0.01i$. The others parameters are the same to those in the case (I). The phase and group refraction angles for the weak and strong chirality are shown in Fig. 5. It can be seen that whatever the chirality is weak or strong, the phase and group refraction angles are always positive (${\theta}_{R}>0,{\theta}_{gR}>0$) for the RCP wave. It indicates that the uniaxial anisotropic chiral metamaterial always supports the forward RCP wave. But for the LCP wave, the phase refraction angles are always positive (${\theta}_{L}>0$), while the group refraction angles are always negative (${\theta}_{gL}<0$). Thus the backward wave can be supported for both the weak and strong chirality, and the uniaxial anisotropic chiral metamaterial possesses negative refraction index. It means that the realization of the negative chiral metamaterial is not affected by the chirality parameter for the case (II). Here, we note that the chirality parameter will affect the negative refractive index much more for the strong chirality, and the negative refractive index will decrease with the increasing of the chirality. Comparing Fig. 2 and Fig. 5, we conclude that the uniaxial anisotropic chiral slab can be negative as the chirality *κ* is strong for case (I), but for case (II) it can be negative whatever the chirality is weak or strong.

Furthermore, the dependence of the reflectivity $|R{|}^{2}$ (a, b) and the GH shifts $\Delta /\lambda $ (c, d) for both perpendicular (subscript sp) and parallel (subscript pp) components on the angle of incidence are discussed. For the perpendicular component, as shown in Fig. 6(a) and 6(c), there is a peak in each reflectivity curve and the GH shifts will be greatly enhanced at close-to-grazing incidence. By comparison, we find that the variation trends of the reflectivity and the GH shifts are the same as those in Fig. 3(a) and 3(c), which means that the influences of the chirality on the perpendicular component are the same for the two cases of the uniaxial chiral slab, i.e., case (I) and case (II). But for the parallel component, there always exists a dip of the reflectivity curve whatever the chirality is weak or strong, at which the reflectivity reaches the minimum magnitude and can be close to zero, the corresponding angle of incidence can be defined as the Brewster angle. As a consequence, the GH shifts are positively enhanced near the Brewster angle, as seen in Fig. 6(d). Meanwhile, with the increasing of chirality, the enhancement of the positive GH shifts decreases (or increases) for the weak (or strong) chirality. Moreover, the Brewster angle shifts to the large angle of incidence with increasing the chirality. As a result, the incident angle corresponding to the enhancement peak shifts to the large angle as the chirality increases. According to Fig. 6, the magnitude and the position of the enhancement of the GH shifts can be adjusted by changing the chirality parameter.

In the end, to show the influences of the thickness of the slab on the GH shifts for the co-polarized reflection. The GH shifts of the parallel components for different thickness are shown in Fig. 7 for different chirality. We can see the GH shifts always have one enhanced peak with the angle of incidence whatever the chirality is weak or strong. Meanwhile, the position of the peak shows a right shift, i.e., the peak shifts to the large incident angle. As chirality is small (see Fig. 7(a)), the enhancement of the GH shifts decreases with the increase of the thickness. While the enhancement of the GH shifts decreases firstly and then increases with increasing the thickness as the chirality is strong, seen in Fig. 7(b).

## 4. Conclusion

In summary, we construct a uniaxial anisotropic chiral metamaterial slab with the optical axis perpendicular to the interface. An investigation on the GH shifts of reflected waves has been done by using the stationary-phase approach. Since there are four cases of electromagnetic parameters for the uniaxial chiral slab, we focus our discussion on the refraction angles and the GH shifts for the two cases of the uniaxial chiral slab, i.e., case (I) (${\epsilon}_{t}>0,{\epsilon}_{z}>0$) and (II) (${\epsilon}_{t}>0,{\epsilon}_{z}<0$). Our numerical results show that for both of the cases, the uniaxial chiral slab can support the backward wave, whose group refraction index of LCP wave is negative. For case (I), the negative uniaxial chiral slab can be realized as the chirality is strong. While, it can be obtained whatever the chirality is weak or strong for case (II). In addition, the variation of the GH shifts of the perpendicular component (${\Delta}_{sp}$) are the same for the two cases of the uniaxial chiral slab, which are both greatly enhanced at close-to-grazing incidence. It indicates that there does not exist the Brewster angle in the cross-polarized reflection, and the influence of the chirality on ${\Delta}_{sp}$ for case (I) and (II) are the same. Meanwhile, For the co-polarized reflection, the Brewster angles are always exist for the two cases, and the chirality will largely affect the GH shifts. For case (I), the GH shifts of the parallel component (${\Delta}_{pp}$) can be negatively or positively enhanced near the Brewster angles. By adjusting the chirality, the enhancements of the GH shifts can transit from two negative peaks to a positive one. For case (II), the GH shifts of the parallel component (${\Delta}_{pp}$) are always positively enhanced near the Brewster angle. By adjusting the chirality, the magnitude of the enhanced GH shifts will decrease, the corresponding position will shift to large incident angle. In the end, whatever the chirality is weak or strong, the transition between negative and positive GH shifts of the parallel component can be realized by adjusting the thickness of the chiral slab for case (I). For case (II), the GH shifts are always positively enhanced and the magnitude of the enhancement of the GH shifts can be tunable by the thickness of the chiral slab.

Some other comments are in the following. We know that the spatial and angle GH shifts occur simultaneously in general. And the angle GH shifts are dominant for a focused incident beam, and then we would like to investigate the angle GH shifts for a focused incident beam in our next work.

## Funding

National Natural Science Foundation of China (NSFC) (11604163, 11447229, 61371057).

## Acknowledgments

We are thankful for the financial support and the fruitful cooperation with all partners.

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